mintOC - User contributions [en]

Goddart's rocket problem (Bocop)

2016-02-01T16:56:55Z

PaulScharnhorst:

This is a [[:Category: Bocop | Bocop]] implementation of [[Goddart's rocket problem]]. The .def, .constants and .bounds files can be obtained through defining the problems dimensions in the GUI. The .tpp files need to be edited separately.
== Problem.def ==
<source lang="text">
# This file defines all dimensions and parameters
# values for your problem :

# Initial and final time :
time.free string final
time.initial double 0
time.final double 1

# Dimensions :
state.dimension integer 3
control.dimension integer 1
algebraic.dimension integer 0
parameter.dimension integer 1
constant.dimension integer 6
boundarycond.dimension integer 4
constraint.dimension integer 1

# Discretization :
discretization.steps integer 100
discretization.method string gauss

# Optimization :
optimization.type string batch
batch.type integer 0
batch.index integer 5
batch.nrange integer 3
batch.lowerbound double 0.3
batch.upperbound double 0.9
batch.directory string Batch-C

# Initialization :
initialization.type string from_init_file
initialization.file string none

# Parameter identification :
paramid.type string false
paramid.separator string ,
paramid.file string no_directory
paramid.dimension integer 0

# Names :
state.0 string position
state.1 string speed
state.2 string mass
control.0 string acceleration_u
parameter.0 string finaltime
boundarycond.0 string r(0)
boundarycond.1 string v(0)
boundarycond.2 string m(0)
boundarycond.3 string r(f)
constraint.0 string drag_minus_C
constant.0 string Tmax
constant.1 string A
constant.2 string k
constant.3 string r0
constant.4 string b
constant.5 string C

# Solution file :
solution.file string problem.sol

# Iteration output frequency :
iteration.output.frequency integer 0
</source>

== Problem.constants ==
<source lang="text">
# This file contains the values of the constants of your problem.
# Number of constants used in your problem :
6

# Values of the constants :
3.5
310
500
1
7
0.6

</source>

== Problem.bounds ==
<source lang="text">
# This file contains all the bounds of your problem.
# Bounds are stored in standard format :
# [lower bound] [upper bound] [type of bound]

# Dimensions (i&f conditions, y, u, z, p, path constraints) :
4 3 1 0 1 1

# Bounds for the initial and final conditions :
1 1 equal
0 0 equal
1 1 equal
1.01 2e+020 lower

# Bounds for the state variables :
>0:1:0 1 2e+020 lower
>1:1:1 0 2e+020 lower
>2:1:2 0 2e+020 lower

# Bounds for the control variables :
0 1 both

# Bounds for the algebraic variables :

# Bounds for the optimization parameters :
0 2e+020 lower

# Bounds for the path constraints :
-2e+020 0 upper
</source>

== criterion.tpp ==
<source lang="cpp">
#include "header_criterion"
{
// CRITERION FOR GODDARD PROBLEM

// MAXIMIZE FINAL MASS
criterion = -final_state[2];
}
</source>

== dynamics.tpp ==
<source lang="cpp">
#include "header_dynamics"
{
// DYNAMICS FOR GODDARD PROBLEM
// dr/dt = v
// dv/dt = (Thrust(u) - Drag(r,v)) / m - grav(r)
// dm/dt = -b*|u|

double Tmax = constants[0];
double A = constants[1];
double k = constants[2];
double r0 = constants[3];
double b = constants[4];

Tdouble r = state[0];
Tdouble v = state[1];
Tdouble m = state[2];

state_dynamics[0] = v;
state_dynamics[1] = (thrust(control[0],Tmax) - drag(r,v,A,k,r0)) / m - grav(r);
state_dynamics[2] = - b * control[0];
}
</source>

== boundarycond.tpp ==
<source lang="cpp">
#include "header_boundarycond"
{
// INITIAL CONDITIONS FOR GODDARD PROBLEM
// r0 = 1 v0 = 0 m0 = 1
// MODELED AS 1 <= r0 <= 1, etc
boundary_conditions[0] = initial_state[0];
boundary_conditions[1] = initial_state[1];
boundary_conditions[2] = initial_state[2];

// FINAL CONDITIONS FOR GODDARD PROBLEM
// rf >= 1.01 MODELED AS 1.01 <= rf
boundary_conditions[3] = final_state[0];
}
</source>

== pathcond.tpp ==
<source lang="cpp">
#include "header_pathcond"
{
// CONSTRAINT ON MAX DRAG FOR GODDARD PROBLEM
// Drag <= C ie Drag - C <= 0

double A = constants[1];
double k = constants[2];
double r0 = constants[3];
double C = constants[5];

Tdouble r = state[0];
Tdouble v = state[1];

path_constraints[0] = drag(r,v,A,k,r0) - C;
}
</source>

== dependencies.tpp ==
The dependencies file is only needed if you use additional functions like the three following.
<source lang="cpp">
#include "./grav.tpp"
#include "./drag.tpp"
#include "./thrust.tpp"
</source>

== drag.tpp ==
<source lang="cpp">
#include "adolc/adolc.h"
#include "adolc/adouble.h"
#include <cmath>

// FUNCTION FOR GODDARD DRAG
// drag = 310 v^2 exp (-500(r-1))

// Arguments:
// r: radius
// v: velocity

template<class Tdouble> Tdouble drag(Tdouble r, Tdouble v, double A, double k, double r0)
{

Tdouble drag;
drag = A * v * v * exp(-k*(fabs(r)-r0));
return drag;

}
</source>

== grav.tpp ==
<source lang="cpp">
#include "adolc/adolc.h"
#include "adolc/adouble.h"
#include <cmath>

// FUNCTION FOR GRAVITY
// g = 1 / r^2

// Arguments:
// r: radius

template<class Tdouble> Tdouble grav(Tdouble r)
{

Tdouble grav;
grav = 1e0 / r / r;
return grav;

}
</source>

== thrust.tpp ==
<source lang="cpp">
#include "adolc/adolc.h"
#include "adolc/adouble.h"
#include <cmath>

// FUNCTION FOR THRUST (GODDARD)
// T = u * Tmax

// Arguments:
// r: radius

template<class Tdouble> Tdouble thrust(Tdouble u, double Tmax)
{
Tdouble thrust;
thrust = u * Tmax;
return thrust;
}

</source>

[[Category: Bocop]]

Goddart's rocket problem (Bocop)

2016-02-01T16:55:26Z

PaulScharnhorst:

This is a [[:Category: Bocop | Bocop]] implementation of [[Goddart's rocket problem]]. The .def, .constants and .bounds files can be obtained through defining the problems dimensions in the GUI. The .tpp files need to be edited separately.
== Problem.def ==
<source lang="text">
# This file defines all dimensions and parameters
# values for your problem :

# Initial and final time :
time.free string final
time.initial double 0
time.final double 1

# Dimensions :
state.dimension integer 3
control.dimension integer 1
algebraic.dimension integer 0
parameter.dimension integer 1
constant.dimension integer 6
boundarycond.dimension integer 4
constraint.dimension integer 1

# Discretization :
discretization.steps integer 100
discretization.method string gauss

# Optimization :
optimization.type string batch
batch.type integer 0
batch.index integer 5
batch.nrange integer 3
batch.lowerbound double 0.3
batch.upperbound double 0.9
batch.directory string Batch-C

# Initialization :
initialization.type string from_init_file
initialization.file string none

# Parameter identification :
paramid.type string false
paramid.separator string ,
paramid.file string no_directory
paramid.dimension integer 0

# Names :
state.0 string position
state.1 string speed
state.2 string mass
control.0 string acceleration_u
parameter.0 string finaltime
boundarycond.0 string r(0)
boundarycond.1 string v(0)
boundarycond.2 string m(0)
boundarycond.3 string r(f)
constraint.0 string drag_minus_C
constant.0 string Tmax
constant.1 string A
constant.2 string k
constant.3 string r0
constant.4 string b
constant.5 string C

# Solution file :
solution.file string problem.sol

# Iteration output frequency :
iteration.output.frequency integer 0
</source>

== Problem.constants ==
<source lang="text">
# This file contains the values of the constants of your problem.
# Number of constants used in your problem :
6

# Values of the constants :
3.5
310
500
1
7
0.6

</source>

== Problem.bounds ==
<source lang="text">
# This file contains all the bounds of your problem.
# Bounds are stored in standard format :
# [lower bound] [upper bound] [type of bound]

# Dimensions (i&f conditions, y, u, z, p, path constraints) :
4 3 1 0 1 1

# Bounds for the initial and final conditions :
1 1 equal
0 0 equal
1 1 equal
1.01 2e+020 lower

# Bounds for the state variables :
>0:1:0 1 2e+020 lower
>1:1:1 0 2e+020 lower
>2:1:2 0 2e+020 lower

# Bounds for the control variables :
0 1 both

# Bounds for the algebraic variables :

# Bounds for the optimization parameters :
0 2e+020 lower

# Bounds for the path constraints :
-2e+020 0 upper
</source>

== criterion.tpp ==
<source lang="cpp">
#include "header_criterion"
{
// CRITERION FOR GODDARD PROBLEM

// MAXIMIZE FINAL MASS
criterion = -final_state[2];
}
</source>

== dynamics.tpp ==
<source lang="cpp">
#include "header_dynamics"
{
// DYNAMICS FOR GODDARD PROBLEM
// dr/dt = v
// dv/dt = (Thrust(u) - Drag(r,v)) / m - grav(r)
// dm/dt = -b*|u|

double Tmax = constants[0];
double A = constants[1];
double k = constants[2];
double r0 = constants[3];
double b = constants[4];

Tdouble r = state[0];
Tdouble v = state[1];
Tdouble m = state[2];

state_dynamics[0] = v;
state_dynamics[1] = (thrust(control[0],Tmax) - drag(r,v,A,k,r0)) / m - grav(r);
state_dynamics[2] = - b * control[0];
}
</source>

== boundarycond.tpp ==
<source lang="cpp">
#include "header_boundarycond"
{
// INITIAL CONDITIONS FOR GODDARD PROBLEM
// r0 = 1 v0 = 0 m0 = 1
// MODELED AS 1 <= r0 <= 1, etc
boundary_conditions[0] = initial_state[0];
boundary_conditions[1] = initial_state[1];
boundary_conditions[2] = initial_state[2];

// FINAL CONDITIONS FOR GODDARD PROBLEM
// rf >= 1.01 MODELED AS 1.01 <= rf
boundary_conditions[3] = final_state[0];
}
</source>

== pathcond.tpp ==
<source lang="cpp">
#include "header_pathcond"
{
// CONSTRAINT ON MAX DRAG FOR GODDARD PROBLEM
// Drag <= C ie Drag - C <= 0

double A = constants[1];
double k = constants[2];
double r0 = constants[3];
double C = constants[5];

Tdouble r = state[0];
Tdouble v = state[1];

path_constraints[0] = drag(r,v,A,k,r0) - C;
}
</source>

== dependencies.tpp ==
<source lang="cpp">
#include "./grav.tpp"
#include "./drag.tpp"
#include "./thrust.tpp"
</source>

== drag.tpp ==
<source lang="cpp">
#include "adolc/adolc.h"
#include "adolc/adouble.h"
#include <cmath>

// FUNCTION FOR GODDARD DRAG
// drag = 310 v^2 exp (-500(r-1))

// Arguments:
// r: radius
// v: velocity

template<class Tdouble> Tdouble drag(Tdouble r, Tdouble v, double A, double k, double r0)
{

Tdouble drag;
drag = A * v * v * exp(-k*(fabs(r)-r0));
return drag;

}
</source>

== grav.tpp ==
<source lang="cpp">
#include "adolc/adolc.h"
#include "adolc/adouble.h"
#include <cmath>

// FUNCTION FOR GRAVITY
// g = 1 / r^2

// Arguments:
// r: radius

template<class Tdouble> Tdouble grav(Tdouble r)
{

Tdouble grav;
grav = 1e0 / r / r;
return grav;

}
</source>

== thrust.tpp ==
<source lang="cpp">
#include "adolc/adolc.h"
#include "adolc/adouble.h"
#include <cmath>

// FUNCTION FOR THRUST (GODDARD)
// T = u * Tmax

// Arguments:
// r: radius

template<class Tdouble> Tdouble thrust(Tdouble u, double Tmax)
{
Tdouble thrust;
thrust = u * Tmax;
return thrust;
}

</source>

[[Category: Bocop]]

Category:Bocop

2016-02-01T16:49:44Z

PaulScharnhorst:

This category lists all problems for which Bocop code is provided.

Bocop stands for 'a toolBox for Optimal COntrol Problems'. It provides an interface for defining optimal control problems and solves these via discretization and optimization of the resulting NLP.

For more information visit the [http://bocop.org/ Bocop] website.

== References ==
<biblist />

[[Category: Implementation]]

Goddart's rocket problem (Bocop)

2016-01-29T15:23:48Z

PaulScharnhorst:

== Problem.def ==
<source lang="text">
# This file defines all dimensions and parameters
# values for your problem :

# Initial and final time :
time.free string final
time.initial double 0
time.final double 1

# Dimensions :
state.dimension integer 3
control.dimension integer 1
algebraic.dimension integer 0
parameter.dimension integer 1
constant.dimension integer 6
boundarycond.dimension integer 4
constraint.dimension integer 1

# Discretization :
discretization.steps integer 100
discretization.method string gauss

# Optimization :
optimization.type string batch
batch.type integer 0
batch.index integer 5
batch.nrange integer 3
batch.lowerbound double 0.3
batch.upperbound double 0.9
batch.directory string Batch-C

# Initialization :
initialization.type string from_init_file
initialization.file string none

# Parameter identification :
paramid.type string false
paramid.separator string ,
paramid.file string no_directory
paramid.dimension integer 0

# Names :
state.0 string position
state.1 string speed
state.2 string mass
control.0 string acceleration_u
parameter.0 string finaltime
boundarycond.0 string r(0)
boundarycond.1 string v(0)
boundarycond.2 string m(0)
boundarycond.3 string r(f)
constraint.0 string drag_minus_C
constant.0 string Tmax
constant.1 string A
constant.2 string k
constant.3 string r0
constant.4 string b
constant.5 string C

# Solution file :
solution.file string problem.sol

# Iteration output frequency :
iteration.output.frequency integer 0
</source>

== Problem.constants ==
<source lang="text">
# This file contains the values of the constants of your problem.
# Number of constants used in your problem :
6

# Values of the constants :
3.5
310
500
1
7
0.6

</source>

== Problem.bounds ==
<source lang="text">
# This file contains all the bounds of your problem.
# Bounds are stored in standard format :
# [lower bound] [upper bound] [type of bound]

# Dimensions (i&f conditions, y, u, z, p, path constraints) :
4 3 1 0 1 1

# Bounds for the initial and final conditions :
1 1 equal
0 0 equal
1 1 equal
1.01 2e+020 lower

# Bounds for the state variables :
>0:1:0 1 2e+020 lower
>1:1:1 0 2e+020 lower
>2:1:2 0 2e+020 lower

# Bounds for the control variables :
0 1 both

# Bounds for the algebraic variables :

# Bounds for the optimization parameters :
0 2e+020 lower

# Bounds for the path constraints :
-2e+020 0 upper
</source>

== criterion.tpp ==
<source lang="cpp">
#include "header_criterion"
{
// CRITERION FOR GODDARD PROBLEM

// MAXIMIZE FINAL MASS
criterion = -final_state[2];
}
</source>

== dynamics.tpp ==
<source lang="cpp">
#include "header_dynamics"
{
// DYNAMICS FOR GODDARD PROBLEM
// dr/dt = v
// dv/dt = (Thrust(u) - Drag(r,v)) / m - grav(r)
// dm/dt = -b*|u|

double Tmax = constants[0];
double A = constants[1];
double k = constants[2];
double r0 = constants[3];
double b = constants[4];

Tdouble r = state[0];
Tdouble v = state[1];
Tdouble m = state[2];

state_dynamics[0] = v;
state_dynamics[1] = (thrust(control[0],Tmax) - drag(r,v,A,k,r0)) / m - grav(r);
state_dynamics[2] = - b * control[0];
}
</source>

== boundarycond.tpp ==
<source lang="cpp">
#include "header_boundarycond"
{
// INITIAL CONDITIONS FOR GODDARD PROBLEM
// r0 = 1 v0 = 0 m0 = 1
// MODELED AS 1 <= r0 <= 1, etc
boundary_conditions[0] = initial_state[0];
boundary_conditions[1] = initial_state[1];
boundary_conditions[2] = initial_state[2];

// FINAL CONDITIONS FOR GODDARD PROBLEM
// rf >= 1.01 MODELED AS 1.01 <= rf
boundary_conditions[3] = final_state[0];
}
</source>

== pathcond.tpp ==
<source lang="cpp">
#include "header_pathcond"
{
// CONSTRAINT ON MAX DRAG FOR GODDARD PROBLEM
// Drag <= C ie Drag - C <= 0

double A = constants[1];
double k = constants[2];
double r0 = constants[3];
double C = constants[5];

Tdouble r = state[0];
Tdouble v = state[1];

path_constraints[0] = drag(r,v,A,k,r0) - C;
}
</source>

== dependencies.tpp ==
<source lang="cpp">
#include "./grav.tpp"
#include "./drag.tpp"
#include "./thrust.tpp"
</source>

== drag.tpp ==
<source lang="cpp">
#include "adolc/adolc.h"
#include "adolc/adouble.h"
#include <cmath>

// FUNCTION FOR GODDARD DRAG
// drag = 310 v^2 exp (-500(r-1))

// Arguments:
// r: radius
// v: velocity

template<class Tdouble> Tdouble drag(Tdouble r, Tdouble v, double A, double k, double r0)
{

Tdouble drag;
drag = A * v * v * exp(-k*(fabs(r)-r0));
return drag;

}
</source>

== grav.tpp ==
<source lang="cpp">
#include "adolc/adolc.h"
#include "adolc/adouble.h"
#include <cmath>

// FUNCTION FOR GRAVITY
// g = 1 / r^2

// Arguments:
// r: radius

template<class Tdouble> Tdouble grav(Tdouble r)
{

Tdouble grav;
grav = 1e0 / r / r;
return grav;

}
</source>

== thrust.tpp ==
<source lang="cpp">
#include "adolc/adolc.h"
#include "adolc/adouble.h"
#include <cmath>

// FUNCTION FOR THRUST (GODDARD)
// T = u * Tmax

// Arguments:
// r: radius

template<class Tdouble> Tdouble thrust(Tdouble u, double Tmax)
{
Tdouble thrust;
thrust = u * Tmax;
return thrust;
}

</source>

[[Category: Bocop]]

Lotka Volterra fishing problem (Bocop)

2016-01-21T09:20:43Z

PaulScharnhorst:

== Problem.def ==
<source lang="text">
# This file defines all dimensions and parameters
# values for your problem :

# Initial and final time :
time.free string none
time.initial double 0
time.final double 12

# Dimensions :
state.dimension integer 3
control.dimension integer 1
algebraic.dimension integer 0
parameter.dimension integer 0
constant.dimension integer 2
boundarycond.dimension integer 3
constraint.dimension integer 0

# Discretization :
discretization.steps integer 100
discretization.method string lobatto

# Optimization :
optimization.type string single
batch.type integer 0
batch.index integer 0
batch.nrange integer 1
batch.lowerbound double 0
batch.upperbound double 0
batch.directory string none

# Initialization :
initialization.type string from_init_file
initialization.file string none

# Parameter identification :
paramid.type string false
paramid.separator string ,
paramid.file string no_directory
paramid.dimension integer 0

# Names :
state.0 string x0
state.1 string x1
state.2 string x2
control.0 string w
boundarycond.0 string x00
boundarycond.1 string x10
boundarycond.2 string x20
constant.0 string c0
constant.1 string c1

# Solution file :
solution.file string problem.sol

# Iteration output frequency :
iteration.output.frequency integer 0
</source>

== Problem.constants ==
<source lang="text">
# This file contains the values of the constants of your problem.
# Number of constants used in your problem :
2

# Values of the constants :
0.4
0.2
</source>

== Problem.bounds ==
<source lang="text">
# This file contains all the bounds of your problem.
# Bounds are stored in standard format :
# [lower bound] [upper bound] [type of bound]

# Dimensions (boundary conditions, state, control, algebraic
# variables, optimization parameters, path constraints) :
3 3 1 0 0 0
0.5 0.5 equal
0.7 0.7 equal
0 0 equal

# Bounds for the initial and final conditions :

# Bounds for the state variables :
>0:1:0 0 0 free
>1:1:1 0 0 free
>2:1:2 0 0 free
0 1 both

# Bounds for the control variables :

# Bounds for the algebraic variables :

# Bounds for the optimization parameters :

# Bounds for the path constraints :
</source>

== criterion.tpp ==
<source lang="cpp">
#include "header_criterion"
{
// HERE : description of the function for the criterion
// "criterion" is a function of all variables X[]
criterion = final_state[2];
}
</source>

== dynamics.tpp ==
<source lang="cpp">
#include "header_dynamics"
{
// HERE : description of the function for the dynamics
// Please give a function or a value for the dynamics of each state variable
double c0 = constants[0];
double c1 = constants[1];

Tdouble x0 = state[0];
Tdouble x1 = state[1];
Tdouble x2 = state[2];

state_dynamics[0] = x0-x0*x1-c0*x0*control[0];
state_dynamics[1] = -x1+x0*x1-c1*x1*control[0];
state_dynamics[2] = (x0-1)*(x0-1)+(x1-1)*(x1-1);
}
</source>

== boundarycond.tpp ==
<source lang="cpp">
#include "header_boundarycond"
{
// HERE : description of the function for the initial and final conditions
// Please give a function or a value for each element of boundaryconditions
boundary_conditions[0] = initial_state[0];
boundary_conditions[1] = initial_state[1];
boundary_conditions[2] = initial_state[2];
}
</source>
[[Category:Bocop]]

Lotka Volterra fishing problem (Bocop)

2016-01-21T09:20:06Z

PaulScharnhorst: Created page with " == Problem.def == <source lang="text"> # This file defines all dimensions and parameters # values for your problem : # Initial and final time : time.free string none time.in..."

Goddart's rocket problem

2016-01-16T12:14:39Z

PaulScharnhorst:

In Goddart's rocket problem we model the ascent (vertical; restricted to 1 dimension) of a rocket. The aim is to reach a certain altitude with minimal fuel consumption. It is equivalent to maximize the mass at the final altitude.
[[Category:MIOCP]]
== Variables ==
The state variables <math>r,v,m</math> describe the altitude(radius), speed and mass.

The drag is given by
<dd>
<math>
D(r,v):= Av^2 \rho(r)\text{, with }\rho(r):= exp(-k\cdot (r-r_0)).
</math>
</dd>
All units are renormalized.

== Mathematical formulation ==

<dd>
<math>
\begin{array}{llcll}
\displaystyle \min_{m,r,v,u,T} & -m(T)\\[1.5ex]
\mbox{s.t.} & \dot{r} & = & v, \\
& \dot{v} & = & -\frac{1}{r^2} + \frac{1}{m} (T_{max}u-D(r,v)) \\[1.5ex]
& \dot{m} & = & -b T_{max} u, \\
& u(\cdot) &\in& [0,1] \\
& r(0) &=& r_0, \\
& v(0) &=& v_0, \\
& m(0) &=& m_0, \\
& r(T) &=& r_T, \\
& D(r(\cdot),v(\cdot))&\le& C \\
& T free
\end{array}
</math>
</dd>

== Parameters ==

<dd>
<math>
\begin{array}{rcl}
r_0 &=& 1 \\
v_0 &=& 0 \\
m_0 &=& 1 \\
r_T &=& 1.01 \\
b &=& 7 \\
T_{max} &=& 3.5 \\
A &=& 310 \\
k &=& 500 \\
C &=& 0.6
\end{array}
</math>
</dd>

== Reference Solution ==

The following reference solution was generated using BOCOP. The optimal value of the objective function is -0.63389.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
Image:Goddartacc.png| Control u over time.
Image:Goddartpos.png| Position r over time.
Image:Goddartspeed.png| Speed v over time.
Image:Goddartmass.png| Mass m over time.
</gallery>

== References ==
The Problem can be found in the [http://bocop.org/ BOCOP User Guide].

Goddart's rocket problem

2016-01-14T13:33:10Z

PaulScharnhorst:

In Goddart's rocket problem we model the ascent (vertical; restricted to 1 dimension) of a rocket. The aim is to reach a certain altitude with minimal fuel consumption. It is equivalent to maximize the mass at the final altitude.
[[Category:MIOCP]]
== Variables ==
The state variables <math>r,v,m</math> describe the altitude, speed and mass.

The drag is given by
<dd>
<math>
D(r,v):= Av^2 \rho(r)\text{, with }\rho(r):= exp(-k\cdot (r-r_0)).
</math>
</dd>
All units are renormalized.

== Mathematical formulation ==

<dd>
<math>
\begin{array}{llcll}
\displaystyle \min_{m,r,v,u,T} & -m(T)\\[1.5ex]
\mbox{s.t.} & \dot{r} & = & v, \\
& \dot{v} & = & -\frac{1}{r^2} + \frac{1}{m} (T_{max}u-D(r,v)) \\[1.5ex]
& \dot{m} & = & -b T_{max} u, \\
& u(\cdot) &\in& [0,1] \\
& r(0) &=& r_0, \\
& v(0) &=& v_0, \\
& m(0) &=& m_0, \\
& r(T) &=& r_T, \\
& D(r(\cdot),v(\cdot))&\le& C \\
& T free
\end{array}
</math>
</dd>

== Parameters ==

<dd>
<math>
\begin{array}{rcl}
r_0 &=& 1 \\
v_0 &=& 0 \\
m_0 &=& 1 \\
r_T &=& 1.01 \\
b &=& 7 \\
T_{max} &=& 3.5 \\
A &=& 310 \\
k &=& 500 \\
C &=& 0.6
\end{array}
</math>
</dd>

== Reference Solution ==

The following reference solution was generated using BOCOP. The optimal value of the objective function is -0.63389.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
Image:Goddartacc.png| Control u over time.
Image:Goddartpos.png| Position r over time.
Image:Goddartspeed.png| Speed v over time.
Image:Goddartmass.png| Mass m over time.
</gallery>

== References ==
The Problem can be found in the [http://bocop.org/ BOCOP User Guide].

Goddart's rocket problem

2016-01-14T13:27:53Z

PaulScharnhorst:

In Goddart's rocket problem we model the ascent (vertical; restricted to 1 dimension) of a rocket. The aim is to reach a certain altitude with minimal fuel consumption. It is equivalent to maximize the mass at the final altitude.
[[Category:MIOCP]]
== Variables ==
The state variables <math>r,v,m</math> describe the altitude, speed and mass.

The drag is given by
<dd>
<math>
D(r,v):= Av^2 \rho(r)\text{, with }\rho(r):= exp(-k\cdot (r-r_0)).
</math>
</dd>
All units are renormalized.

== Mathematical formulation ==

<dd>
<math>
\begin{array}{llcll}
\displaystyle \min_{m,r,v,u,T} & -m(T)\\[1.5ex]
\mbox{s.t.} & \dot{r} & = & v, \\
& \dot{v} & = & -\frac{1}{r^2} + \frac{1}{m} (T_{max}u-D(r,v)) \\[1.5ex]
& \dot{m} & = & -b T_{max} u, \\
& u(\cdot) &\in& [0,1] \\
& r(0) &=& r_0, \\
& v(0) &=& v_0, \\
& m(0) &=& m_0, \\
& r(T) &=& r_T, \\
& D(r(\cdot),v(\cdot))&\le& C \\
& T free
\end{array}
</math>
</dd>

== Parameters ==

<dd>
<math>
\begin{array}{rcl}
r_0 &=& 1 \\
v_0 &=& 0 \\
m_0 &=& 1 \\
r_T &=& 1.01 \\
b &=& 7 \\
T_{max} &=& 3.5 \\
A &=& 310 \\
k &=& 500 \\
C &=& 0.6
\end{array}
</math>
</dd>

== Reference Solution ==

The following reference solution was generated using BOCOP. The optimal value of the objective function is -0.63389.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
Image:Goddartacc.png| Control u over time.
Image:Goddartpos.png| Position r over time.
Image:Goddartspeed.png| Speed v over time.
Image:Goddartmass.png| Mass m over time.
</gallery>

Goddart's rocket problem

2016-01-14T13:27:23Z

PaulScharnhorst:

In Goddart's rocket problem we model the ascent (vertical; restricted to 1 dimension) of a rocket. The aim is to reach a certain altitude with minimal fuel consumption. It is equivalent to maximize the mass at the final altitude.
[[Category:MIOCP]]
== Variables ==
The state variables <math>r,v,m</math> describe the altitude, speed and mass.

The drag is given by
<dd>
<math>
D(r,v):= Av^2 \rho(r)\text{, with }\rho(r):= exp(-k\cdot (r-r_0)).
</math>
</dd>
All units are renormalized.

== Mathematical formulation ==

<dd>
<math>
\begin{array}{llcll}
\displaystyle \max_{m,r,v,u,T} & m(T)\\[1.5ex]
\mbox{s.t.} & \dot{r} & = & v, \\
& \dot{v} & = & -\frac{1}{r^2} + \frac{1}{m} (T_{max}u-D(r,v)) \\[1.5ex]
& \dot{m} & = & -b T_{max} u, \\
& u(\cdot) &\in& [0,1] \\
& r(0) &=& r_0, \\
& v(0) &=& v_0, \\
& m(0) &=& m_0, \\
& r(T) &=& r_T, \\
& D(r(\cdot),v(\cdot))&\le& C \\
& T free
\end{array}
</math>
</dd>

== Parameters ==

<dd>
<math>
\begin{array}{rcl}
r_0 &=& 1 \\
v_0 &=& 0 \\
m_0 &=& 1 \\
r_T &=& 1.01 \\
b &=& 7 \\
T_{max} &=& 3.5 \\
A &=& 310 \\
k &=& 500 \\
C &=& 0.6
\end{array}
</math>
</dd>

== Reference Solution ==

The following reference solution was generated using BOCOP. The optimal value of the objective function is -0.63389.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
Image:Goddartacc.png| Control u over time.
Image:Goddartpos.png| Position r over time.
Image:Goddartspeed.png| Speed v over time.
Image:Goddartmass.png| Mass m over time.
</gallery>

File:Goddartspeed.png

2016-01-14T13:21:46Z

PaulScharnhorst:

File:Goddartpos.png

2016-01-14T13:21:32Z

PaulScharnhorst:

File:Goddartacc.png

2016-01-14T13:21:12Z

PaulScharnhorst:

File:Goddartmass.png

2016-01-14T13:20:57Z

PaulScharnhorst:

Goddart's rocket problem

2016-01-14T13:20:08Z

PaulScharnhorst:

Goddart's rocket problem

2016-01-11T12:53:32Z

PaulScharnhorst: Created page with "In Goddart's rocket problem we model the ascent (vertical; restricted to 1 dimension) of a rocket. The aim is to reach a certain altitude with minimal fuel consumption. It is..."